real number system 2 (sup, inf)

In preceding post (real number system), we accepted the axiom for completeness [41] on real number system. For discussing this issue, some definitions or propositions will be required.

A set $M$ of real numbers is bounded above if there is an upper bound $a$ such that if $m\in M$, $m\leq a$.

As real number system is an ordered field, it is clear that if $M$ is bounded above, there is a lot of upper bound $a$. That is, there is a set of upper bound numbers, or a set by which $M$ is bounded above.

Similarly, a set $N$ of real numbers is bounded below if there is a lower bound $b$ such that if $n\in N$, $n\geq b$.

Of course, there is a set of lower bound numbers.

If $\alpha$ is an upper bound of $M$, but there is not any upper bound number less than $\alpha$, then $\alpha$ is the least upper bound number of $M$ and is called the supremum of $M$, and we write
$$\alpha = \sup M$$
If $\beta$ is the greatest lower bound number of $N$ and the infimum of $N$,
$$\beta = \inf N$$

You may recall $\max M$ or $\min N$. However, $\max M$ must be an element of $M$ and $\min N$ must be an element of $N$, too. Let us note that $\sup M$ may be an element of $M$, or may not be, and $\inf N$ may be an element of $N$, or may not be.